Integral Calculus Notes

Indefinite Integrals and U-Substitution

Basic Integral Formulas

• $\int \sec(x) \tan(x) dx = \sec(x) + C$
• $\int \csc(x) \cot(x) dx = -\csc(x) + C$

Inverse Trigonometric Integrals

• $\int \frac{dx}{\sqrt{1 - x^2}} = \sin^{-1}(x) + C$
• $\int \frac{-dx}{\sqrt{1 - x^2}} = \cos^{-1}(x) + C$

Power Rule Integrals

• The general power rule for integration:
  $\int [g(x)]^n g'(x) dx = \frac{[g(x)]^{n+1}}{n+1} + C$ where $n \neq -1$
• If $g(x) = x$, this simplifies to the basic power rule:
  $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

U-Substitution

• U-substitution is useful when the integrand contains a composite function $f(g(x))$ and its derivative $g'(x)$. The goal is to transform the integral into a simpler form by substituting $u = g(x)$ and $du = g'(x) dx$.

Examples

Example 1: Power Rule with Trigonometric Function

• Evaluate $\int \sin^2(x) \cos(x) dx$
• Here, $g(x) = \sin(x)$, and $g'(x) = \cos(x)$.
• Applying the power rule:
  $\int \sin^2(x) \cos(x) dx = \frac{\sin^3(x)}{3} + C$

Example 2: Power Rule with Polynomial Function

• Evaluate $\int (2x^2 + 3x)^2 (4x + 3) dx$
• Here, $g(x) = 2x^2 + 3x$, and $g'(x) = 4x + 3$.
• Applying the power rule:
  $\int (2x^2 + 3x)^2 (4x + 3) dx = \frac{(2x^2 + 3x)^3}{3} + C$

Example 3: U-Substitution with Exponential Function

• Evaluate $\int e^{x^2} x dx$
• Let $u = x^2$, then $du = 2x dx$, and $x dx = \frac{1}{2} du$
• The integral becomes:
  $\int e^{x^2} x dx = \frac{1}{2} \int e^u du = \frac{1}{2} e^u + C = \frac{1}{2} e^{x^2} + C$

Explanation of Exponential Integral

• The integral $\int e^{g(x)} g'(x) dx = e^{g(x)} + C$
• Consider the derivative of $F(x) = e^{x^2} + C$:
  $\frac{dF}{dx} = \frac{d}{dx}(e^{x^2} + C) = e^{x^2} (2x)$
• This verifies that $e^{x^2}$ is the antiderivative of $e^{x^2} (2x)$.

Example 4: U-Substitution with Inverse Sine

• Evaluate $\int \frac{2x}{\sqrt{1 - (x^2)^2}} dx$
• Let $g(x) = x^2$, so the integral resembles the form $\int \frac{g'(x)}{\sqrt{1 - [g(x)]^2}} dx$
• Therefore, $\int \frac{2x}{\sqrt{1 - (x^2)^2}} dx = \sin^{-1}(x^2) + C$

Example 5: U-Substitution with Inverse Tangent

• Evaluate $\int \frac{2x}{1 + x^4} dx$
• Rewrite the integral as $\int \frac{2x}{1 + (x^2)^2} dx$
• Let $g(x) = x^2$, then $g'(x) = 2x$
• Using the formula $\int \frac{g'(x)}{1 + [g(x)]^2} dx = \tan^{-1}(g(x)) + C$:
  $\int \frac{2x}{1 + x^4} dx = \tan^{-1}(x^2) + C$

General Cases

• Type 1: Integrals where $g(x) = x$ is a specific (simpler) case.

• Type 2: Integrals involving a more general function $g(x)$.

• $\int \frac{dx}{x \sqrt{x^2 - 1}} = \sec^{-1}(|x|) + C$

• For a general case, it becomes $\int \frac{g'(x)}{g(x) \sqrt{[g(x)]^2 - 1}} dx = \sec^{-1}(|g(x)|) + C$

Example 6: Inverse Tangent with Trigonometric Function

• Evaluate $\int \frac{\cos(x)}{1 + \sin^2(x)} dx$
• Let $g(x) = \sin(x)$, so $g'(x) = \cos(x)$
• Using the formula $\int \frac{g'(x)}{1 + [g(x)]^2} dx = \tan^{-1}(g(x)) + C$:
  $\int \frac{\cos(x)}{1 + \sin^2(x)} dx = \tan^{-1}(\sin(x)) + C$

Example 7: Inverse Sine with Exponential Function

• Evaluate $\int \frac{7^x \ln(7)}{\sqrt{1 - (7^x)^2}} dx$
• Let $g(x) = 7^x$, so $g'(x) = 7^x \ln(7)$
• Using the formula $\int \frac{g'(x)}{\sqrt{1 - [g(x)]^2}} dx = \sin^{-1}(g(x)) + C$:
  $\int \frac{7^x \ln(7)}{\sqrt{1 - (7^x)^2}} dx = \sin^{-1}(7^x) + C$

Example 8: Logarithmic Integral

• Evaluate $\int \frac{1}{1 + x} dx$
• Recognize this as $\int \frac{g'(x)}{g(x)} dx = \ln|g(x)| + C$:
  $\int \frac{1}{1 + x} dx = \ln|1 + x| + C$